Materials analysis using X-rays provides accurate data in a number of applications and industries. X-ray fluorescence measurements allow the determination of the elemental composition of a sample. In some applications however this is not enough and there is a need not merely to determine the elemental composition but also to determine structure parameters such as the crystalline phases of a sample and X-ray diffraction is used in these cases.
Typically, high resolution X-ray diffraction measurements are carried out in reflection mode, where an incoming beam of X-rays is incident on a first surface of a sample and the X-rays diffracted by a diffraction angle 2θ from the same surface of the sample are detected by a detector.
In some applications it is useful to be able to take X-ray diffraction measurements in a transmission mode, where the X-rays are incident on a first surface of a sample and diffracted by a diffraction angle 2θ are measured after passing through the sample from the first surface to the opposite second surface.
A problem with making measurements in this transmission geometry is that the sample itself may be absorbing for X-rays. Therefore, it is difficult to carry out accurate quantitative analysis of the diffracted X-rays to determine the amount of any given phase of the sample, since the absorption of X-rays in the sample is not in general known.
There is therefore a need for a method that quantitatively carries out a matrix and thickness correction for X-ray diffraction.
Absorption of electromagnetic waves that pass directly through a medium is characterised by the Beer-Lambert lawI=I0e−μρd where I0 is the original intensity, I the intensity after passing through the material, μ the mass attenuation coefficient of the material, ρ the material density and d the material thickness (i.e. the ray path length in the material).
The calculation of the effects of absorption in quantitative X-ray measurements is made more difficult than this simple formula would suggest for a number of reasons.
In the simple Beer-Lambert case where X-rays pass directly through a sample without deviation it is possible to characterise the effect of the absorption on the measured X-ray intensity simply by a single value, the value of the product μρd. This is not possible where the X-rays of interest are diffracting X-rays or otherwise redirected and to accurately characterise the absorption requires two parameters, the product μρd as well as the mass absorption coefficient μ.
In this regard, some samples have constraints on thickness. For a pressed powder sample, a suitable thickness of the sample that will result in a sample that is sufficiently strong to be handled and measured will be at least 2 mm, preferably 3 mm. However, at these thicknesses, for typical X-ray energies required for many applications, the absorption of X-rays in the thickness of a sample is higher than 50%. This means that the effects of absorption are large and any deviances from the simple Beer-Lambert law significant. A large absorption means that the relationship between measured intensity and concentration of a particular component of a sample is not straightforward.
A further problem is that the absorption is a function of the composition of the sample. Small changes in the concentration of various components in the sample can cause significant changes in absorption. This is a problem for quantitative X-ray analysis designed to measure the quantity of a given component in the sample, since the amount of that component is unknown but will affect the absorption.
A yet further problem when measuring pressed powder samples is that the thickness d is not generally exactly known. In general, in an industrial environment, it will be desired to make a pressed powder sample and then measure it as soon as possible. It is generally undesirable to have to make accurate measurements of thickness d before carrying out X-ray measurement.
These difficulties may be seen with reference to FIG. 1 which illustrates the theoretically calculated diffraction intensity for free lime as a function of sample thickness for three samples of standard cement clinker materials (Portland cement clinker) mixed with a wax binder for various binder percentages of 0%, 10%, 20% and 30%. Note that in spite of the fact that the samples of higher thickness contain more diffracting material—a sample of twice the thickness has twice the amount of free lime—the diffracted intensity is in fact less. Realistic sample thicknesses around 3 mm are in the highly non-linear regime in which there is no simple relationship between measured intensity and amount of free lime in the sample.
Further, as illustrated in FIG. 2, the diffraction intensity is also dependent on the exact composition. FIG. 2 shows three graphs for three different samples each of Portland cement clinker. In spite of the general similarity between the samples, the diffraction intensity still varies from sample to sample illustrating that the effect of absorption is a function of the exact composition which varies from sample to sample. At a thickness of 3 mm a difference of about 8% in diffraction intensity is seen. This too makes calculating a quantitative measure of free lime concentration from diffraction measurements difficult.
The effects of a variable composition on quantitative measurement is known as a matrix correction since it depends on the composition of the measured sample, i.e. the matrix. It is in general difficult to calculate the matrix correction. There is therefore a need for a measurement method which avoids this difficulty.